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Limit comparison test

Let a n , b n 0 for all n 1 .

  1. If lim n a n / b n = L 0 , then n = 1 a n and n = 1 b n both converge or both diverge.
  2. If lim n a n / b n = 0 and n = 1 b n converges, then n = 1 a n converges.
  3. If lim n a n / b n = and n = 1 b n diverges, then n = 1 a n diverges.

Note that if a n / b n 0 and n = 1 b n diverges, the limit comparison test gives no information. Similarly, if a n / b n and n = 1 b n converges, the test also provides no information. For example, consider the two series n = 1 1 / n and n = 1 1 / n 2 . These series are both p -series with p = 1 / 2 and p = 2 , respectively. Since p = 1 / 2 > 1 , the series n = 1 1 / n diverges. On the other hand, since p = 2 < 1 , the series n = 1 1 / n 2 converges. However, suppose we attempted to apply the limit comparison test, using the convergent p series n = 1 1 / n 3 as our comparison series. First, we see that

1 / n 1 / n 3 = n 3 n = n 5 / 2 as n .

Similarly, we see that

1 / n 2 1 / n 3 = n as n .

Therefore, if a n / b n when n = 1 b n converges, we do not gain any information on the convergence or divergence of n = 1 a n .

Using the limit comparison test

For each of the following series, use the limit comparison test to determine whether the series converges or diverges. If the test does not apply, say so.

  1. n = 1 1 n + 1
  2. n = 1 2 n + 1 3 n
  3. n = 1 ln ( n ) n 2
  1. Compare this series to n = 1 1 n . Calculate
    lim n 1 / ( n + 1 ) 1 / n = lim n n n + 1 = lim n 1 / n 1 + 1 / n = 1 .
    By the limit comparison test, since n = 1 1 n diverges, then n = 1 1 n + 1 diverges.
  2. Compare this series to n = 1 ( 2 3 ) n . We see that
    lim n ( 2 n + 1 ) / 3 n 2 n / 3 n = lim n 2 n + 1 3 n · 3 n 2 n = lim n 2 n + 1 2 n = lim n [ 1 + ( 1 2 ) n ] = 1 .

    Therefore,
    lim n ( 2 n + 1 ) / 3 n 2 n / 3 n = 1 .

    Since n = 1 ( 2 3 ) n converges, we conclude that n = 1 2 n + 1 3 n converges.
  3. Since ln n < n , compare with n = 1 1 n . We see that
    lim n ln n / n 2 1 / n = lim n ln n n 2 · n 1 = lim n ln n n .

    In order to evaluate lim n ln n / n , evaluate the limit as x of the real-valued function ln ( x ) / x . These two limits are equal, and making this change allows us to use L’Hôpital’s rule. We obtain
    lim x ln x x = lim x 1 x = 0 .

    Therefore, lim n ln n / n = 0 , and, consequently,
    lim n ln n / n 2 1 / n = 0 .

    Since the limit is 0 but n = 1 1 n diverges, the limit comparison test does not provide any information.
    Compare with n = 1 1 n 2 instead. In this case,
    lim n ln n / n 2 1 / n 2 = lim n ln n n 2 · n 2 1 = lim n ln n = .

    Since the limit is but n = 1 1 n 2 converges, the test still does not provide any information.
    So now we try a series between the two we already tried. Choosing the series n = 1 1 n 3 / 2 , we see that
    lim n ln n / n 2 1 / n 3 / 2 = lim n ln n n 2 · n 3 / 2 1 = lim n ln n n .

    As above, in order to evaluate lim n ln n / n , evaluate the limit as x of the real-valued function ln x / x . Using L’Hôpital’s rule,
    lim x ln x x = lim x 2 x x = lim x 2 x = 0 .

    Since the limit is 0 and n = 1 1 n 3 / 2 converges, we can conclude that n = 1 ln n n 2 converges.
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Use the limit comparison test to determine whether the series n = 1 5 n 3 n + 2 converges or diverges.

The series diverges.

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Key concepts

  • The comparison tests are used to determine convergence or divergence of series with positive terms.
  • When using the comparison tests, a series n = 1 a n is often compared to a geometric or p -series.

Use the comparison test to determine whether the following series converge.

n = 1 a n where a n = 2 n ( n + 1 )

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n = 1 a n where a n = 1 n ( n + 1 / 2 )

Converges by comparison with 1 / n 2 .

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul
Practice Key Terms 2

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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